Sparse approximation of multilinear problems with applications to kernel-based methods in UQ

Sparse approximation of multilinear problems with applications to kernel-based methods in UQ

​ Soeren Wolfers, Fabio Nobile, Raul Tempone, Sparse approximation of multilinear problems with applications to kernel-based methods in UQ. Submitted arXiv 1609.00246, Sept 2016.
Soeren Wolfers, Fabio Nobile, Raul Tempone
Sparse approximation of multilinear problems with applications to kernel-based methods in UQ
2016
We provide a framework for the sparse approximation of multilinear problems
and show that several problems in uncertainty quantification fit within this
framework. In these problems, the value of a multilinear map has to be
approximated using approximations of different accuracy and computational work
of the arguments of this map. We propose and analyze a generalized version of
Smolyak's algorithm, which provides sparse approximation formulas with
convergence rates that mitigate the curse of dimension that appears in
multilinear approximation problems with a large number of arguments. We apply
the general framework to response surface approximation and optimization under
uncertainty for parametric partial differential equations using kernel-based
approximation. The theoretical results are supplemented by numerical
experiments.